# Law of the iterated logarithm

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2010 Mathematics Subject Classification: Primary: 60F10 Secondary: 60F15 [MSN][ZBL]

A limit theorem in probability theory which is a refinement of the strong law of large numbers. Let $X _ {1} , X _ {2} \dots$ be a sequence of random variables and let

$$S _ {n} = X _ {1} + \dots + X _ {n} .$$

For simplicity one assumes that $S _ {n}$ has zero median for each $n$. While the theorem on the strong law of large numbers deals with conditions under which $S _ {n} /a _ {n} \rightarrow 0$ almost surely ( $a.s.$) for $n \rightarrow \infty$, where $\{ a _ {n} \}$ is a sequence of numbers, the theorem on the law of the iterated logarithm deals with sequences of numbers $\{ c _ {n} \}$ such that

$$\tag{1 } \lim\limits _ {n \rightarrow \infty } \sup \frac{S _ {n} }{c _ {n} } = 1 \ ( a.s.)$$

or

$$\tag{2 } \lim\limits _ {n \rightarrow \infty } \sup \frac{| S _ {n} | }{c _ {n} } = 1 \ ( a.s.).$$

Relation (1) is equivalent to

$${\mathsf P} \{ S _ {n} > ( 1 + \epsilon ) c _ {n} ( i.o.) \} = 0$$

and

$${\mathsf P} \{ S _ {n} > ( 1 - \epsilon ) c _ {n} ( i.o.) \} = 1$$

for any $\epsilon > 0$, where $i.o.$ denotes infinitely often.

Relations of the form of (1) and (2) hold under more restrictive conditions than the estimates implied by the strong law of large numbers. If $\{ X _ {n} \}$ is a sequence of independent random variables having identical distributions with mathematical expectations equal to zero, then

$$\frac{S _ {n} }{n} \rightarrow 0 \ \textrm{ (a.s.) } \ \textrm{ for } n \rightarrow \infty$$

(Kolmogorov's theorem); if the additional condition $0 < {\mathsf E} X _ {1} ^ {2} < \infty$ is satisfied, then one has the stronger relation (2), in which

$$c _ {n} = ( 2nb { \mathop{\rm ln} \mathop{\rm ln} } ( nb)) ^ {1/2} ,$$

where $b = {\mathsf E} X _ {1} ^ {2}$( the Hartman–Wintner theorem).

The first theorem of general type on the law of the iterated logarithm was the following result obtained by A.N. Kolmogorov [Ko]. Let $\{ X _ {n} \}$ be a sequence of independent random variables with mathematical expectations equal to zero and with finite variances, and let

$$B _ {n} = \sum _ { k= } 1 ^ { n } {\mathsf E} X _ {k} ^ {2} .$$

If $B _ {n} \rightarrow \infty$ for $n \rightarrow \infty$ and if there exists a sequence of positive constants $\{ M _ {n} \}$ such that

$$| X _ {n} | \leq M _ {n} \ \textrm{ and } \ \ M _ {n} = o \left ( \left ( \frac{B _ {n} }{ \mathop{\rm ln} \mathop{\rm ln} B _ {n} } \right ) ^ {1/2} \right ) ,$$

then (1) and (2) are satisfied for

$$c _ {n} = ( 2B _ {n} \mathop{\rm ln} \mathop{\rm ln} B _ {n} ) ^ {1/2} .$$

In the particular case where $\{ X _ {n} \}$ is a sequence of independent random variables having identical distributions with two possible values, this assertion was derived by A.Ya. Khinchin [Kh]. J. Marcinkiewicz and A. Zygmund [MZ] showed that under the conditions of Kolmogorov's theorem one cannot replace $o$ by $O$. W. Feller [F] examined a generalization of Kolmogorov's law of the iterated logarithm for sequences of independent bounded non-identically distributed random variables. See [S] for other generalizations of the law; there is also the following result (see [S2]), which is related to the Hartman–Wintner theorem: If $\{ X _ {n} \}$ is a sequence of independent random variables having identical distributions with infinite variances, then

$$\lim\limits _ {n \rightarrow \infty } \sup \frac{| S _ {n} | }{( n \mathop{\rm ln} n \mathop{\rm ln} n) ^ {1/2} } = \infty \ ( a.s.).$$

The results obtained on the law of the iterated logarithm for sequences of independent random variables have served as a starting point for numerous researches on the applicability of this law to sequences of dependent random variables and vectors and to random processes.

#### References

 [Ko] A.N. Kolmogoroff, "Ueber das Gesetz des iterierten Logarithmus" Math. Ann. , 101 (1929) pp. 126–135 [Kh] A. Khintchine, "Ueber einen Satz der Wahrscheinlichkeitsrechnung" Fund. Math. , 6 (1924) pp. 9–20 Zbl 50.0344.02 [MZ] J. Marcinkiewicz, A. Zygmund, "Rémarque sur la loi du logarithme itéré" Fund. Math. , 29 (1937) pp. 215–222 Zbl 0018.03204 Zbl 63.1076.03 [F] W. Feller, "The general form of the so-called law of the iterated logarithm" Trans. Amer. Math. Soc. , 54 (1943) pp. 373–402 MR0009263 Zbl 0063.08417 [S] V. Strassen, "An invariance principle for the law of the iterated logarithm" Z. Wahrsch. Verw. Geb. , 3 (1964) pp. 211–226 MR0175194 Zbl 0132.12903 [S2] V. Strassen, "A converse to the law of iterated logarithm" Z. Wahrsch. Verw. Geb. , 4 (1965–1966) pp. 265–268 Zbl 0141.16501 [HW] P. Hartman, A. Wintner, "On the law of the iterated logarithm" Amer. J. Math. , 63 (1941) pp. 169–176 MR0003497 Zbl 0024.15802 Zbl 67.0460.03 [La] J. Lamperty, "Probability" , Benjamin (1966) [P] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042